| L(s) = 1 | + 2.04·3-s + 1.18·9-s − 3.56·11-s + 2.90·13-s + 3.71·17-s + 3.57·19-s + 3.10·23-s − 3.70·27-s + 1.57·29-s + 0.764·31-s − 7.29·33-s + 7.00·37-s + 5.95·39-s − 7.07·41-s + 10.9·43-s − 7.37·47-s + 7.61·51-s − 3.32·53-s + 7.32·57-s − 8.14·59-s + 5.93·61-s + 14.0·67-s + 6.36·69-s − 14.5·71-s + 13.2·73-s + 2.98·79-s − 11.1·81-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.396·9-s − 1.07·11-s + 0.806·13-s + 0.901·17-s + 0.821·19-s + 0.648·23-s − 0.713·27-s + 0.293·29-s + 0.137·31-s − 1.26·33-s + 1.15·37-s + 0.952·39-s − 1.10·41-s + 1.66·43-s − 1.07·47-s + 1.06·51-s − 0.457·53-s + 0.970·57-s − 1.05·59-s + 0.760·61-s + 1.71·67-s + 0.765·69-s − 1.72·71-s + 1.54·73-s + 0.335·79-s − 1.23·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.343531443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.343531443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 0.764T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.37T + 47T^{2} \) |
| 53 | \( 1 + 3.32T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78792748007171954993922187981, −7.28844642864198849886579072001, −6.30480337529526587331827864438, −5.58277555315588893695765024285, −4.93689517705097679784448519292, −3.98104132951090848252145992075, −3.17199126456231421806438618923, −2.85063157285975723535059293386, −1.86582047644801137167793717549, −0.828340952966374943516942758400,
0.828340952966374943516942758400, 1.86582047644801137167793717549, 2.85063157285975723535059293386, 3.17199126456231421806438618923, 3.98104132951090848252145992075, 4.93689517705097679784448519292, 5.58277555315588893695765024285, 6.30480337529526587331827864438, 7.28844642864198849886579072001, 7.78792748007171954993922187981
